Verify that f satisfies the conditions of the mean-value...

Verify that $f$ satisfies the conditions of the mean-value theorem on the indicated interval and find all numbers $c$ that satisfy the conclusion of the theorem.
$$f(x)=\sqrt{1-x^{2}} ; \quad[0,1]$$

Key Concepts

Mean Value Theorem

The Mean Value Theorem is a fundamental concept in calculus which asserts that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point within the open interval where the derivative of the function (its instantaneous rate of change) equals the average rate of change over the entire interval. This theorem provides a formal bridge between the behavior of the function over an interval and its behavior at individual points.

Continuity and Differentiability Conditions

Ensuring that a function is continuous on a closed interval and differentiable on the corresponding open interval is essential for applying the Mean Value Theorem. Continuity on the closed interval guarantees there are no breaks or jumps in the graph of the function, while differentiability on the open interval means that the function has a well-defined tangent at every point within that interval. These conditions collectively secure the prerequisites needed for the theorem to hold.

Derivative Calculation

Calculating the derivative of a function is key in using the Mean Value Theorem, as the theorem involves setting the derivative equal to the function’s average rate of change on the interval. This process utilizes differentiation rules, such as the chain rule, to accurately determine the instantaneous rate of change at any point in the interior of the interval.

Equating the Derivative to the Average Rate of Change

After determining the derivative of the function, the next step involves computing the average rate of change of the function over the interval, which is done by subtracting the function's values at the endpoints and dividing by the length of the interval. Setting the derivative equal to this average rate of change and solving for the critical point(s) c is how one finds the specific value(s) that satisfy the conclusion of the Mean Value Theorem.

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